Lesson_18 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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EEL 3135: Dr. Fred J. Taylor, Professor Lesson Title: Convolution Theorem Lesson Number: 18 [Sections 7-3 to 7-5 with part of 7-6.] Background: It has been establish that the z-operator represents a simple delay. Specifically, z -k represents k clocked delays where each delay has a duration T s =1/f s . Furthermore, it was established that the z-transform of a discrete-time signal modeled as: [ ] [ ] [ ] k n k x n x N k - = = δ 0 1. is: ( 29 [ ] k N k z k x z X - = = 0 2 The z-transform is a linear operator, therefore superposition holds. In particular, if x 1 [n] and x 2 [n] are discrete-time signals, and x[n]= α x 1 [n] + β x 2 [n], then: x[n]= α x 1 [n] + β x 2 [n] α X 1 (z) + β X 2 (z) = X(z) 3 The delay property states that if X(z) is the z-transform of a signal x[n], then the z-transform of x[n-k] is given by: ( 29 : ] [ z X z k n x k Z - → - 4. The concept of a transfer function was developed in Lesson 17, where: ( 29 [ ] n h z h z H Z k N k k = - = 0 5 The values of z which set H(z) to zero (zero gain) are called the filter’s zeros . Example Design a 15 th order lowpass FIR with a passband cutoff frequency equal to f 0 = 0.1f s . Display the zero locations and FIR’s frequency response. >> h=fir1(14,0.2); >> zplane(h,1) 1
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EEL 3135: Dr. Fred J. Taylor, Professor Figure 1: Magnitude frequency response of a 15 th order lowpass FIR (top) and zero locations (bottom). Convolution The output of a discrete-time LTI filter impulse response h [ k ], to an input x [ k ], was formally given in Lesson 10 to be defined by the convolution sum: [ ] [ ] = - = = 0 m m] h[m]x[k y[k] k x k h 6 Suppose now that both h [ k ] and x [ k ] possess known z -transforms given by: X(z) x[k] H(z) h[k] → Z Z 7 The Z -transform of Equation 5 is then formally given by: 2
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EEL 3135: Dr. Fred J. Taylor, Professor = - - = = = = = = 0 m 0 = k k k 0 k 0 = m 0 = m m]z - x[k h[m] z m] - x[k h[m] m] - x[k h[m] x[k]] [h[k] [y[k]] Y(z) Z Z Z 8 Let p=k-m , it then follows that: ( 29 H(z)X(z) x[p]z h[m]z x[p]z h[m] [y[k]] Y(z) 0 m 0 p= p -m 0 m 0 p= m p+ = = = = = = - = - Z 9 Equation 9 is also known by its popular name, the convolution theorem for z -transforms. It can be seen that the z -transform of the convolution sum of products operation y [ k ]= h [ k ] x [ k ] is mathematically equivalent to multiplying the z -transforms of h [ k ] and x [ k ], producing the output transform Y [ z ] in the z -domain. Once in the z -transform of the output (Y(z)) is known, it can be transformed back into the discrete-time domain usin a technique that will be called an inverse z- transform (subject of a future study) This theorem provides a bridge between time-domain convolution and transform operations.
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_18 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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